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This note announces three density theorems involving representations of Lie algebras and associative algebras. The first theorem describes the irreducible (possibly infinite dimensional) representations p of a Lie algebra g with an ideal ï such that the restriction of p to ï has some absolutely irreducible quotient representation. The second result is an embedding theorem for the irreducible representations of the Weyl algebras A niC over C (A n>c^C [t l9 • • • , t n , d/d^, • • • , 9/9^J, thedoi:10.1090/s0002-9904-1974-13547-0 fatcat:25rju7d5jbb6dojeacslwamdwe